The terminology elliptical for the billiards we shall deal with, refers to the shape of the billiard table. (See e.g. [1]}, Ch. 4.) This has not to be confounded with hyperbolic billiards, for which we refer to [2]}. It seems worth mentioning that several remarkable investigations in the topic of billiards regard especially polygonal billiards, which indeed occur among the most natural first examples (see e.g. [1]}, Ch. 7, for generalities). These are related in particular to the geometry of suitable curves of genus at least 2 and their Jacobians (see e.g. W. Veech's paper [4]} and C. McMullen's papers [5]}, [6]}, and references therein). While we do not see any definite direct connection of our billiards with the polygonal ones, it is puzzling that both our context and the latter have links with torsion divisors in Jacobians and the Pell-Abel equationWe follow J-P. Serre's Séminaire Bourbaki paper [7]} for this terminology., i.e. a polynomial version of the famous Pell equation in Number Theory.
